All of the above literature on differential privacy requires a decomposable system objective function \(f_t\left( \mathbf {x} \right)\) , i.e., \(f_t\left( \mathbf {x} \right) =\sum \nolimits _{i=1}^n{f_{i,t}\left( \mathbf {x} \right)}\) , in which any local objective \(f_{i,t}\)  is just revealed to node \(i\)  at time \(t\) . This case is described as overlap [1]}, in which each node \(i\)  holds and updates an estimate \(\mathbf {x}_i\in \mathbb {R}^d\)  that converges asymptotically to the best decision \(\mathbf {x}^{\ast }\in \mathbb {R}^d\)  of the system problem. Beware that the dimension \(d\)  is possibly consistent with the number of nodes \(n\)  or not. However, some objective functions \(f_t\left( \mathbf {x} \right)\)  may not allow decomposition operations, and the decision variable \(\mathbf {x}=\left( x_1,\cdots ,x_n \right)^{\top } \in \mathbb {R}^n\)  is stored fragmentarily in all the distributed nodes, where each node \(i\)  only controls a part \(x_i\)  of \(\mathbf {x}\) . This case is described as specialization [1]}.
